Today we will
discuss strain energy stored in a body when load will be applied gradually with
the help of this post.

Let us
consider a body which is subjected with tensile load which is increasing
gradually up to its elastic limit from value 0 to value P and therefore
deformation or extension of the body is also increasing from 0 to x and we can
see it in following load extension diagram as displayed here.

We have
following information from above load extension diagram for body which is subjected
with tensile load up to its elastic limit.

σ= Stress
developed in the body

E = Young’s
Modulus of elasticity of the material of the body

A= Cross
sectional area of the body

P =
Gradually applied load which is increasing gradually up to its elastic limit from
value 0 to value P

P = σ. A

x =
Deformation or extension of the body which is also increasing from 0 to x

L = Length
of the body

V= Volume
of the body = L.A

U = Strain
energy stored in the body

As we have
already discussed that when a body will be loaded within its elastic limit, the
work done by the load in deforming the body will be equal to the strain energy
stored in the body.

Strain
energy stored in the body = Work done by the load in deforming the body

Strain
energy stored in the body = Area of the load extension curve

Strain
energy stored in the body = Area of the triangle ABC

U = (1/2) . AB . BC

U = (1/2)
x. P

Let us use
the value of P = σ. A, which is determined above

U = (1/2)
x. σ. A

Now we
will secure the value of extension x in terms of Stress, Length of the body and
Young’s modulus of the body by using the concept of Hook’s Law.

Within elastic limit, stress applied over an elastic
material will be directionally proportional to the strain produced due to
external loading and mathematically we can write above law as mentioned here.

Stress = E. Strain

Where E is Young’s Modulus of elasticity of the material

σ
= E. ε

σ
= E. (x/L)

x = σ.
L/ E

Let use the value of the extension or deformation “x”
in strain energy equation and we will have

U = (1/2)
(σ.
L/ E).σ. A

U = (1/2)
(σ2/E)
L.A

U = (σ2/2E) V

U = (σ2/2E) V

Therefore strain
energy stored in a body, when load will be applied gradually, will be given by following
equation.

Proof resilience

Proof
resilience is basically defined as the maximum strain energy stored in the
body. As we know that strain energy stored in the body will be maximum when
body will be loaded up to its elastic limit, therefore if σ could be taken at
the elastic limit then we will have following equation for Proof resilience.

Modulus of resilience

Modulus of
resilience is a property of the material which is defined as the proof
resilience per unit volume of the body and we can express as